A characterization of best φ-approximants with applications to multidimensional isotonic approximation

Some properties of best monotone approximants in several variables are obtained. We prove the following abstract characterization theorem. Let (Omega, A, ) be a measurable space and let L subset of A be a sigma-lattice. If f belongs to a Musielak-Orlicz space L-phi(Omega, A, mu), then there exists a sigma-algebra A(f) subset of A such that g is a best phi-approximant to f from L-phi(L) iff g is a best phi-approximant to f from L-phi(A(f)). The sigma-algebra A(f) depends only on f. When Omega subset of R-n and L-phi(L) is the set of monotone functions in several variables, we give sufficient conditions on the geometry of Omega to obtain a uniqueness theorem. This result extends and unifies previous ones. Finally, we prove a coincidence relation between a function and its best phi-approximant. Our main results are new, even in the classical Lebesgue spaces L-p.